Bayesian inference of nuclear-matter density from proton scattering

Background: Proton elastic scattering at intermediate energy is widely employed as a tool for determining the matter radius of atomic nuclei. The sensitivity of the approach relies on high-resolution measurements at small scattering angles and low-momentum transfer. Under these conditions, the Glauber multiple scattering theory accurately describes the proton-nucleus elastic cross section.Purpose: Investigate the sensitivity of the Glauber multiple scattering theory to uncertainties associated with input parameters such as the nuclear-matter density distribution and nucleon-nucleon data.Method: A joint Bayesian inference was performed using 12 angular distributions of elastic scattering at different energies on $^{58}\mathrm{Ni}$, $^{90}\mathrm{Zr}$, and $^{208}\mathrm{Pb}$ targets. A Metropolis-Hastings algorithm was implemented to make an uncertainty quantification analysis for the input parameters used in the Glauber multiple scattering theory.Results: The experimental cross sections were fitted simultaneously using a joint Bayesian inference approach. Posterior probability density distributions of 42 input parameters were obtained from the analysis. A moderate correlation between the nuclear density parameters and the nucleon-nucleon cross sections was found. This correlation impacts the extraction of the nuclear-matter radius.Conclusions: The present analysis provided a consistent method for extracting the nuclear-matter density distribution of $^{58}\mathrm{Ni}$, $^{90}\mathrm{Zr}$, and $^{208}\mathrm{Pb}$ from data across different incident energies. Due to the correlation of the nucleon-nucleon cross sections with the other input parameters, a constrained Bayesian inference using free nucleon-nucleon cross section data was performed. The nuclear-matter radii obtained from the analysis are in good agreement with multiple results reported in the literature.